On a problem of Nagata related to Zariski's problem
\textit{M. Nagata} [in Osaka J. Math. 33, No. 4, 997-1002 (1996; see the preceding review Zbl 0944.13012)] has introduced the notion of divisorial valuation ring. Namely, let \(B\) be a noetherian (integral) domain. We say that \(D\) is a divisorial valuation ring over \(B\) if \(D\) is some localization of the derived normal ring (i.e. the integral closure in its field of quotients) of an affine ring over \(B\) by a height one prime ideal. Related to his work on Zariski's problem (see op.cit.), he proved that If \(B\) is a noetherian integral domain for which the following property holds: (*) Let \(K\) be the quotient field of \(B\). Then for every divisorial valuation ring \(D\) over \(B\), the intersection of \(D\) with \(K\) is a divisorial valuation ring over \(B\), or is equal to \(K\). Then a ring \(R\) over \(B\) is the intersection of the integral closure of an affine ring over \(B\) with a function field over \(B\), if and only if \(R\) is the \(I\)-transform of the integral closure of some affine ring over \(B\). Motivated by this result he asked the question: What is the class of noetherian integral domains for which the condition (*) above holds ? In the paper under review, the author shows (theorem 2.3), using localization and properties of unibranched local domains, that the property (*) holds for every noetherian domain, giving a simpler restatement of Nagata's results.
scientific article; zbMATH DE number 1217815